\section{Paradigm Insurer}
\label{sec:ParadigmInsurer}
	
The Paradigm Insurer (PI), randomly selects 1,000,000 policyholders (Loss ratios), collects premiums, pays losses and, at year end, calculates its estimate (PLRE) of the population loss ratio (PLR). We assume that PI's standard error, $s_{1,000,000}$, is 0.05 and will show below that we can calculate the value for the standard error for any portfolio size using the CLT (28) and $s_{1,000,000}$.
	
We will use portfolio adjusted standard errors and our assumption that PLREs are normally distributed, to analyze and compare the performance of PI and larger and smaller insurers, before and after PI shifts from fee for service to PCIR. We summarize PI's performance in Exhibit 1 Column 4 but in more detail, PI:

\begin{itemize}
\item Earns premiums of \$4,000,000,000 (\$4,000 per year, per policyholder)
\item Has underwriting and non-loss related expenses of \$600,000,000 (Expense ratio = 0.15)
\item Has a profit Provision = \$200,000,000 (0.05 * \$4,000,000,000)
\item Risk Premium = \$200,000,000 (0.05 * \$4,000,000,000)
\item P[Losses $\leq$ \$3,000,000,000 = P[Profit $\geq$ 10\%] = 0.5000
\item P[Losses $\leq$ \$3,200,000,000 = P[Profit $\geq$ 5\%] = 0.8413
\item P[Losses $\leq$ \$3,400,000,000 = P[Avoid losses] = 0.9772
\item P[Losses $\geq$ \$3,600,000,000 = P[Avoid losses $\geq$ 5\%] = 0.9987
\item P[Losses $\geq$ \$3,800,000,000 = P[Avoid losses $\geq$ 10\%] = 1.0000
\end{itemize}	
	
The probabilities we have assigned to PI's outcomes are calculated by evaluating the cumulative probabilities for PI's normally distributed PLREs (N[0.75, 0.05]), for loss ratios at, or below, 0, 1, 2, 3, and 4 standard errors above the PLR. These evaluation points are equivalent to PLREs of: 0.7500; 0.8000; 0.8500; 0.9000 and 0.9500, respectively. 
